The Problem with Problems

I’m currently reading “Burn Math Class,” and it’s got me thinking about language. Yesterday, I saw an item about teaching students why cancelling works in this case: \[5 + 3 – 3\] but not in this case: \[5 + 3 = 5 – 3\] The conclusion that the students were led to is that the equal sign makes it into two different problems.

I winced, but didn’t say anything in that conversation. There seems to be a significant chunk of mathematics education devoted to narratives like this: “The answer to \(5 + 3 – 3\) is \(5\).” No, no, no. Mathematics doesn’t consist of finding “answers” (usually single numbers) to “problems” (usually multiple expressions combined with operators). That conversation didn’t feel like the place to rant, again, about how that’s not what mathematics is really about.

When used properly, the equality sign separates two expressions that are mathematically equivalent. As far as proper mathematics goes, there’s no sense of one being the “answer” and the other being the “problem”. One may be simpler than the other, or easier to see as a numeric value, or whatever.

The goal of an exercise might be to write an expression in a “simpler” form, although what qualifies as “simpler” may even differ from what you might expect. For instance, \(5\sqrt{2}\) is usually considered “simpler” than \(\sqrt{50}\), even though it looks more complex.

The goal of an exercise might be to find the specific value or values of a variable that render an equation true. For instance, \(4x + 5 = 13\) is only true when \(x\) has the value of 2. Since we speak of all the values that satisfy an equation as collectively being “the solution”, it’s mathematically true to say that \(x = \{2\}\) is the solution for \(4x + 5 = 13\). We often casually say \(x = 2\) is the solution for \(4x + 5 = 13\), but by this we do not truly mean that \(4x + 5 = 13\) is the “problem” and \(x = 5\) is the “answer” in the more casual senses of these words.

When we persist in using words like “problem” and “answer”, we perpetuate the notion that mathematics is about finding a single expression that is the “simplest” form of some value. Usually, the goal in algebra problems is to make a confusing situation more manageable. This might be through finding a solution set, but it might also be through rearranging a relationship to focus on a target variable.

We might want to know the formula for converting from degrees Celsius to degrees Fahrenheit. The “answer” to this “problem” is not a single value, it’s an equation. In this case, it’s \[F = \frac{9C}{5} + 32\]

Neither of these formulas is inherently superior to the other one. For that matter, we could write: \[5F – 9C = 160\]

Mathematically speaking, these three equations are identical: They are true for exactly the same set of ordered real number pairs \((F, C)\). Our choice of one over the others depends completely on what we intend to use it for.

When we focus on finding “answers” to “problems,” I think we miss this.